To find the height of the apartment building, we can use the Pythagorean theorem. The Pythagorean theorem states that for a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Here’s how you can determine the height step-by-step:
1. Identify the sides of the right triangle:
- The hypotenuse is the distance from the tip of the shadow to the top of the building, which is 100 meters.
- One leg of the triangle is the shadow length, which is 72 meters.
- The other leg of the triangle is the height of the building, which we need to find.
2. Use the Pythagorean theorem:
The Pythagorean theorem can be expressed as:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
where [tex]\(c\)[/tex] is the hypotenuse, and [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the legs of the triangle.
3. Rearrange to solve for the unknown height [tex]\(h\)[/tex]:
In this case, [tex]\(b\)[/tex] is the shadow length (72 meters), and [tex]\(c\)[/tex] is the hypotenuse (100 meters). [tex]\(a\)[/tex] is the height of the building.
- Rearrange the formula to solve for [tex]\(a\)[/tex]:
[tex]\[
a^2 = c^2 - b^2
\][/tex]
4. Substitute the known values:
[tex]\[
h^2 = 100^2 - 72^2
\][/tex]
- Calculate [tex]\(100^2\)[/tex]:
[tex]\[
100^2 = 10000
\][/tex]
- Calculate [tex]\(72^2\)[/tex]:
[tex]\[
72^2 = 5184
\][/tex]
- Subtract the two values:
[tex]\[
h^2 = 10000 - 5184 = 4816
\][/tex]
5. Find the square root:
Finally, take the square root of the result to find the height of the building:
[tex]\[
h = \sqrt{4816}
\][/tex]
- Rounding to one decimal place, we get:
[tex]\[
h \approx 69.4 \text{ meters}
\][/tex]
Therefore, the height of the building is approximately 69.4 meters.
